A NOTE ON ACTIONS OF A MONOIDAL CATEGORY
G. JANELIDZE AND G.M. KELLY
ABSTRACT. An action ∗ : V × A−→ A of a monoidal category V on a category A corresponds to a strong monoidal functor F : V−→[A,A]into the monoidal category of endofunctors of A. In many practical cases, the ordinary functor f : V−→[A,A] underlying the monoidalF has a right adjointg; and when this is so,F itself has a right adjoint G as a monoidal functor—so that, passing to the categories of monoids (also called “algebras”) inV and in [A,A], we have an adjunction MonF ⊣MonGbetween the category MonV of monoids inV and the category Mon[A,A] = MndA of monads onA. We give sufficient conditions for the existence of the right adjointg, which involve the existence of right adjoints for the functors X∗ – and – ∗A, and make A (at least when V is symmetric and closed) into a tensored and cotensored V-category A. We give explicit formulae, as large ends, for the right adjoints g and MonG, and also for some related right adjoints, when they exist; as well as another explicit expression for MonGas a large limit, which uses a new representation of any monad as a (large) limit of monads of two special kinds, and an analogous result for general endofunctors.
1. Introduction
Recall from [EK] that a monoidal functor F : V−→ V′ between monoidal categories
V = (V,⊗, I) and V′ = (V′,⊗′, I′) consists ofa triple F = (f,f , f ◦) where f : V−→ V′
is an ordinary functor, fis a natural transformation with components fXY : f X ⊗′f Y
−→f(X⊗Y), and f◦ : I′−→f I is a morphism in V′, these data being required to
satisfy three coherence conditions, corresponding respectively to the associativities, the left identities, and the right identities of ⊗ and of ⊗′. Recall further that a monoidal
natural transformation η from F = (f,f , f ◦) : V−→ V′ to G = (g,g, g◦) : V−→ V′ is
just an ordinary natural transformation η : f−→g : V−→ V′ satisfying two coherence
conditions—one for ⊗ and one for I; and that monoidal categories, monoidal functors, and monoidal natural transformations constitute a 2-categoryMON CAT with a forgetful 2-functorU : MON CAT −→ CAT sending (V,⊗, I) to the underlying category V and so on. Recall finally that a monoidal functor F = (f,f , f ◦) is said to be strong when f◦
and all the fXY are invertible.
Here we are usingMON CAT andCAT to denote the appropriate “meta-2-categories” in which the structure ofmonoidal categories or ofcategories is codified, along with the corresponding morphisms and 2-cells, while size is absent from our thoughts: so that one cannot think ofthe objects ofCAT as forming a set. Similarly we have the meta-category
The authors acknowledge with gratitude the support of the Australian Research Council, which has made it possible for them to work together for several extended periods in Sydney and elsewhere.
Received by the editors 2000 November 24 and, in revised form, 2001 December 4. Published on 2001 December 15 in the volume of articles from CT2000.
2000 Mathematics Subject Classification: 18C15, 18D10, 18D20.
Key words and phrases: monoidal category, action, enriched category, monoid, monad, adjunction. c
G. Janelidze and G.M. Kelly, 2001. Permission to copy for private use granted.
SET ofsets. However we also have a notion ofsmall set, determined by an inaccessible cardinal∞, and hence notions ofsmall category and ofsmall monoidal category; we write Set for the category of small sets, which is the full subcategory of SET with the small sets as objects, and similarly Cat and MonCat for the 2-categories of small categories and ofsmall monoidal categories. Ofcourse any honest category (that is, one whose morphisms form a set) can be rendered small, by a suitable choice of ∞: for we suppose every cardinal to be exceeded by some inaccessible one.
By an action ofa monoidal category V = (V,⊗, I) on a category A we mean a strong monoidal functor F = (f,f , f ◦) : V−→ [A,A], where [A,A] is the category of
endofunctors of A, provided with the (strict) monoidal structure ([A, A],◦,1A), wherein
◦ denotes composition and 1A is the identity endofunctor. Here, to give the functor
f :V−→ [A,A] is equally to give a functor∗:V × A−→ AwhereX∗A = (f X)A; to give the invertible and natural fXY : (f X)◦(f Y)−→f(X⊗Y) (or rather their inverses) is
equally to give a natural isomorphism with componentsαXY A : (X⊗Y)∗A−→X∗(Y∗A);
to give the invertible f◦ : 1
A−→f I (or rather its inverse) is equally to give a natural
isomorphism with components λA : I ∗A−→A; and the coherence conditions for F
become the commutativity ofthe three diagrams
((X⊗Y)⊗Z)∗A α //
wherein a, ℓ, r are the associativity, left identity, and right identity isomorphisms for the monoidal structure on V. Thus an action of V on A can equally be described by giving the functor∗:V × A−→A along with the natural isomorphismsα and λsatisfying (1.1) – (1.3). {In fact, (1.2) is a consequence of(1.1) and (1.3) : the proofofthe corresponding result for a monoidal category given by Kelly in [KM] extends unchanged to the bicategory case, while the data (V,⊗, I, a, ℓ, r,∗, α, λ) satisfying the monoidal-category axioms and (1.1) – (1.3) can be seen as describing a two-object bicategory.}
the corresponding f : V−→ [V,V] has (f X)Y = X ⊗Y. For another simple example ofan action, take V to be the cartesian closed category Set and A to be any category admitting small multiples (also called copowers—for X ∈ Set and A ∈ A, the multiple X·Ais the coproduct of X copies of A); clearly we have an action of SetonA for which X∗A=X·A.
Actions in the sense above occur widely in mathematics; and we have moreover no-ticed that, in many important cases, the functor f : V−→[A,A] admits a right adjoint g : [A,A]−→ V. When this is so, the situation is richer than one might at first think. To begin with, because the monoidal functor F is strong, it follows from Theorem 1.5 ofKelly’s [KD] that this monoidal F : V−→[A,A] has a right adjoint G : [A,A]−→ V
in the 2-category MON CAT exactly when the ordinary functor f : V−→[A,A] has a right adjointg : [A,A]−→ V inCAT (that is, a right adjoint in the classical sense). More precisely, given an adjunction η, ǫ:f ⊣g, there are a uniquegand a unique g◦ for which
(g,g, g◦) is a monoidal functor G : [A,A]−→ V and for which η, ǫ constitute an
adjunc-tionF ⊣GinMON CAT. Next, there is a 2-functor Mon :MON CAT −→ CAT taking a monoidal category V to the category MonV of monoids in V (also known by many writers as algebras in V). Such a monoid is an object M of V together with morphisms m:M ⊗M−→M and i:I−→M satisfying the evident associativity and unit axioms— and is effectively the same thing as a monoidal functor M = (M, m, i) : 1−→ V, where 1 is the unit category with its unique monoidal structure. So Mon : MON CAT −→ CAT
may be seen as the representable 2-functor MON CAT(1,−); and like any 2-functor, it takes adjunctions to adjunctions. In particular, for an action F : V−→ [A,A] of V on
A, an adjunction η, ǫ : f ⊣ g in CAT, being in effect the same thing as an adjunction η, ǫ:F ⊣G inMON CAT, gives us in CAT an adjunction
Monη, Monǫ : MonF ⊣ MonG : MonV −→ MndA, (1.4)
where MndA is the category ofmonads on A and monad-morphisms, which is another name for the category Mon[A,A] of monoids in [A,A] and monoid-maps. Ofcourse the underlying object ofthe monoid (MonF)(M, m, i) is just f M and so on; speaking informally, one says that a monoidal functor takes monoids to monoids.
Because ofthese consequences, it is ofinterest to investigate conditions on an action F :V−→[A,A] under which f :V−→ [A,A] has a right adjointg; and we do this below. In Section 2 we show that anecessary condition forf to have a right adjoint is that each
small with a small generating set, we show that f will have a right adjoint when, to the the necessary condition above that each − ∗A:V−→ Ahave a right adjoint, we add the further condition that each X∗ − :A−→ Ahave a right adjoint. We go on to observe in addition that, at least in the simplest case whereV is symmetric monoidal closed, to give an action ofV onAfor which both − ∗Aand X∗ −have right adjoints is equivalently to give aV-categoryAthat is both tensored and cotensored. In fact, under these conditions, there is a right adjoint not only for f :V−→ [A,A] but also for the appropriate functor q from V to the category (A,A) of V-functors A−→A and V-natural transformations between these. We consider in particular (both here and later) the special cases given by
A=V and by V =Set. In Section 4 we treat the expressions for the right adjoint g of f and the right adjoint r of q as the ends gT =A∈AA(A, T A) andrH =A∈AA(A, HA);
because these arelarge ends, we can use them only when we already know—for example, using the sufficient conditions ofSection 3—that gT and rH exist. Finally, in Section 5, we give a different expression for gT as a large limit, using a new result expressing any monad as a canonical limit ofmonads oftwo special kinds, with a corresponding result for mere endofunctors. In addition, there are scattered throughout the sections a number ofexamples and ofcounter-examples.
2. An argument from the evaluation functors
Our concern being primarily with categories that occur in normal mathematical discourse, we have no reluctance about accepting hypotheses requiring V or A to be complete, cocomplete, or locally small; in many cases ofinterest, all ofthese will hold.
When A is locally small and admits (small) powers—and so certainly if A is locally small and complete—the evaluation functor e(A) : [A,A]−→ A sending T : A−→ A to T A has a right adjoint sending B ∈ A to the power BA(−,A). In that case a necessary condition for f : V−→ [A,A] to have a right adjoint is that each e(A)f have a right adjoint. Since e(A)f is the functor − ∗A : V−→ A, this is to require that each − ∗A have a right adjoint—given, say, by a natural isomorphism
κ:A(X∗A, B) ∼= V(X,A(A, B)). (2.1)
For the rest ofthe paper, we shall suppose that we do have an adjunction (2.1), whether or notA is locally small and admits powers; we lose little ofvalue by restricting ourselves to actions with this property. When A is V itself, with the canonical action given by X∗A=X⊗A, one often writes (2.1) in the form
π :V(X⊗A, B) ∼= V(X,[A, B]), (2.2)
using [A, B] rather than V(A, B); and then we say that the monoidal category V is right closed.
(canonically isomorphic to) A. In fact the composition operation
M :A(B, C)⊗A(A, B)−→A(A, C)
is the morphism corresponding under (an instance of) the adjunction (2.1) to the com-posite
(A(B, C)⊗A(A, B))∗A α //A(B, C)∗(A(A, B)∗A)
1∗ǫAB
//A(B, C)∗B
ǫBC //C , (2.3)
where ǫAB : A(A, B)∗A−→B is the B-component ofthe counit ǫA ofthe adjunction
(2.1); the unit operation j :I−→A(A, A) corresponds under (2.1) to λ:I∗A−→A; the associativity and unit axioms follow just as in the special case A=V, treated in Section 1.6 of[KB] or, in greater detail, in Sections II.3 and II.4 of[EK]; and the isomorphism between A and A0 is given by the isomorphisms
A(A, B)
A(λ,1) //A(I∗A, B) κ //V(I,A(A, B)) =A0(A, B). (2.4) It is moreover straightforward to verify that, ifAis identified withA0by this isomorphism, the functorA:Aop× A−→ V occurring in the adjunction (2.1) coincides with the functor homA :Aop0 ×A0−→ V arising (as in Theorem I.8.2 of[EK]) from theV-categoryA(and described more simply, when V is symmetric, in Section 1.6 of[KB]).
In particular, ofcourse, whenV is right closed, it is itself(isomorphic to) the underlying category V0 ofa V-category V having V(X, Y) = [X, Y] as its “internal hom”.
When the V acting onA and admitting the adjunction (2.1) is in fact right closed, we can say more about the V-categoryA. By Yoneda, there is a unique natural transforma-tion
k=kXAB :A(X∗A, B)−→[X,A(A, B)] (2.5)
making commutative the diagram
A(Y ∗(X∗A), B) κ //
A(α,B)
V(Y,A(X∗A, B))
V(Y,k)
A((Y ⊗X)∗A, B) κ //V(Y ⊗X,A(A, B))
π //V(Y,[X,A(A, B)]) ;
(2.6)
and in f actkis invertible since, by the invertibility ofα, κ,andπ, eachV(Y, k) is invertible. Recall now that we have the representable V-functors A(X∗A,−) : A−→V, A(A,−) : A−→V, and [X,−] :V−→V, as well as the compositeV-functor [X,A(A,−)] :A−→V.
2.1. Lemma. ThekXAB of (2.5) is equal to the composite
A(X∗A, B) A
(A,−)X∗A,B
//[A(A, X∗A),A(A, B)]
[δ,1] //[X,A(A, B)], (2.7)
where δ=δAX :X−→A(A, X∗A) is the X-component of the unit δA of the adjunction
(2.1). It follows from this that kXAB is the B-component of a V-natural isomorphism
Proof. Since A(A,−)X∗A,B corresponds under the adjunction (2.2) to the composition
wherein the top region commutes by naturality, the triangle by one ofthe triangular equations for the adjunction (2.1), and the bottom region by the definition of M; that is, we are to show that κ−1π−1(k) =ǫα. This, however, is exactly what we find when we set Y = A(X ∗A, B) in (2.6) and evaluate both legs at 1 ∈ V(Y,A(X∗A, B)). So k has the form (2.7); and now theV-naturality inB of kXAB follows from Proposition I.8.3
and I.8.4 of[EK]. (Alternatively, thekXA of(2.8) is the unique V-natural transformation
corresponding, by the enriched Yoneda Lemma, to the morphism δ :X−→A(A, X∗A) of V.)
The existence ofthe V-enriched representation (2.8) of[X,A(A,−)] is expressed by saying that the V-category A is tensored, with X ∗A as the tensor product of X and A. (Although this notion oftensor product was originally introduced only for the case of a symmetric monoidal closed V, it clearly makes sense for any right-closed monoidal V; indeed Gordon and Power [GP] extend the definition to the case where V is replaced by a right-closed bicategory.)
2-cells as well, to provide an equivalence of2-categories; and they do so with a general right-closed bicategory in place ofour monoidal V. We have two reasons for giving, nev-ertheless, a briefaccount ofour own : first, things are simpler in our more restricted situation, and with our more restricted goals; and secondly, the treatment in [GP] leaves rather a lot to the reader—such as the proofofour Lemma 2.1 above, covered only by their remark on page 181 (where our V is their W) that “It is routine to verify that this structure does form a tensored W-category”.
Our Introduction contains two examples ofsuch tensored V-categories : first, the canonical action of V on itselfcorresponds, when V is right closed, to the tensored V -category V—and for later use, we shall denote the isomorphism k of(2.5) in this special case by
p: [X⊗Y, Z]−→[X,[Y, Z]]; (2.9)
secondly, Set is certainly right closed, and a tensored Set-category is just an ordinary locally-small category admitting small multiples.
3. Use ofthe Special Adjoint Functor Theorem
We continue to suppose that we have an action F : V−→[A,A], given either by F = (f,f , f ◦) or, alternatively, by ∗ : V × A−→ A along with α and λ, for which we have
the adjunction (2.1) for each pair (X, A); and we now consider what more may suffice to ensure a right adjoint for f :V−→[A,A].
Recall that e(A) : [A,A]−→ A is the functor given by evaluation at A ∈ A. The compositee(A)f :V−→ A, which is − ∗A, is a left adjoint by (2.1), and hence preserves whatever colimits exist in V; that is, a colimit cone qj : Xj−→Y in V gives for each A
a colimit cone qj ∗A : Xj ∗A−→Y ∗A in A. Since diagrams in [A,A] admit colimits
formed pointwise when their evaluations admit colimits in A, such a colimit cone qj inV
gives a colimit conef qj :f Xj−→f Y in [A,A]; in other words, f :V−→ [A,A] preserves
whatever colimits exist in V. To say that f has a right adjoint g is to say that, for each T ∈[A,A], the functor h= [A,A](f−, T) :Vop−→ SET is representable. Note that this functor h takes its values in SET—indeed in some legitimate full subcategory SET of
SET—but not in general in the categorySet ofsmall sets; for [A,A] need not be locally small even when A is so. Observe that h preserves all limits that exist in Vop, since f preserves colimits. So, by the Special Adjoint Functor Theorem as given in Ch.V, Section 8 of Mac Lane’s [ML], h is representable if
(i) V is cocomplete and locally small;
(ii) V admits all cointersections (even large ones, ifneed be) ofepimorphisms;
(iii) there is a small subset ofthe objects ofV that constitutes a generator;
(%) h:Vop−→ SET takes its values in the full subcategory Set ofsmall sets.
Note that (ii) is implied by (i) if V is cowellpowered; and that one cannot omit the condition (%), even though it is often left implicit when the theorem is stated. We shall now show, however, that (%) is a consequence ofthe following three conditions:
(iv) A is locally small;
(v) there is a small subset ofthe objects ofA that constitutes a generator;
and
(vi) each f X :A−→ Ais a left adjoint.
To establish (%), we are to show that each [A,A](f X, T) is a small set, whereX ∈ V
and T :A−→ A. Write S for f X, which by (vi) is a left adjoint. Let the small subsetK
ofthe set ofobjects ofA be the generator given by (v); to say that it is a generator is to say that, for eachAinA, the familyP ofall morphismsp:K−→Awith codomainAand with domainK =K(p) in K is jointly epimorphic. Because S :A−→ Ais a left adjoint, the family given by the Sp:SK(p)−→SA for p in P is again jointly epimorphic. For a functor T : A−→ A and a natural transformation α : S−→T, therefore, it follows from the naturality squareαA.Sp=T p.αKthat the function [A,A](S, T)−→ΠK∈KA(SK, T K)
sending α to{αK|K ∈ K}is injective; which shows [A,A](S, T) to be a small set.
So the six conditions (i) – (vi) suffice for the existence of a right adjoint g for f. The first five ofthese conditions bear only on the categories V and A; note that, by a separate application ofthe Special Adjoint Functor Theorem, (i), (ii), and (iii) imply that V is complete, and similarly (iv) and (v) imply that A is complete whenever it is cocomplete and admits all cointersections ofepimorphisms. Now let us re-cast (vi), which is a condition on the action itself, expressing it in terms of the functor ∗: V × A−→ A. Since f X =X∗ −, condition (vi) is the requirement that, in addition to the adjunction
− ∗A ⊣ A(A,−) of(2.1), we also have for eachX ∈ V an adjunction
A(X∗A, B) ∼= A(A,|X, B|); (3.1)
then, ofcourse, |, | is a functor Vop× A−→ A.
When A isV itself, with∗=⊗, (3.1) takes the form
V(X⊗Y, Z) ∼= V(Y,[[X, Z]]); (3.2)
a monoidal V with such an adjunction is ofcourse said to be left closed. A monoidal V
admitting both the adjunctions (2.2) and (3.2) was formerly said to be biclosed, but is now more commonly said to be simply closed. Recall that Vrev is the monoidal category which isV as a category, with the same unit objectI, but which has the “reverse” tensor product ⊗rev given by
ofcourseVrev is right closed when V is left closed, and conversely. When the monoidal V is braided, we have an isomorphismVrev ∼=V ofmonoidal categories, which is involutory when V is actually symmetric; in these cases a right closed V is also left closed, and thus closed.
Recall too that each V-category A gives rise to a Vrev-category A∗, which has the
same objects as A but has
A∗(A, B) = A(B, A); (3.4)
the ordinary category (A∗)0 underlying A∗ is ofcourse (A0)op. When V is braided, the isomorphism Vrev ∼= V takes us f rom the Vrev-category A∗ to a V-category called Aop, having Aop(A, B) = A(B, A) and (Aop)0 = (A0)op; but when V is only braided and not symmetric, the canonical isomorphism A∼= (Aop)op ceases to be an identity.
In the presence ofthe adjunction (2.1), to ask for the existence ofan adjunction (3.1) is equally to ask for the existence of an adjunction
A(A,|X, B|) ∼= V(X,A(A, B)). (3.5)
Since we can rewrite this as
Aop(|X, B|, A) ∼
= Vrev(X,A∗(B, A)), (3.6)
it is equivalent, whenVis left closed, to the assertion that theVrev-categoryA∗is tensored.
In the case ofa braided closed V, and in particular ofa symmetric closed V, this is equally to say that the V-category Aop is tensored, or equivalently that the V-category A is cotensored, in the sense that we have for each X and B an isomorphism
A(A,|X, B|) ∼= [X,A(A, B)] (3.7)
which is V-natural in A.
3.1. Theorem. Let the monoidal V be cocomplete and locally small, admit all
cointer-sections of epimorphisms, and have a small generating set (and so be complete as well). Let the action F = (f,f , f ◦) : V−→[A,A] of V on the category A correspond as above
to the functor ∗ : V × A−→ A. Then for f, and hence F and MonF, to admit a right adjoint, it suffices that A be locally small with a small generating set, and that − ∗A and X ∗ − admit right adjoints as in (2.1) and (3.1). When the monoidal V is also closed, to give a category A and an action of V on A with the properties above is equally to give a tensored V-category A for which A∗ is tensored and for which the underlying ordinary category A0 has a small generating set. When the monoidal closed V is braided, and in particular when it is symmetric, we can replace “A∗ is tensored” in the preceding sentence
by “A is cotensored”; such an A is in fact complete and cocomplete if A0 is cocomplete and admits all cointersections of epimorphisms.
In particular, we have as a special case:
3.2. Corollary. Let the monoidal closed V be cocomplete and locally small, admit all
cointersections of epimorphisms, and have a small generating set (and hence be complete as well); then the f : V−→[V,V] sending X to f X = X⊗– has a right adjoint g, while the MonF : MonV −→ Mon[V,V] = MndV sending a monoid M in V to the monad M⊗– on V has a right adjoint MonG sending the monad (T, m, i) to a monoid whose underlying object is gT.
3.3. Remark. There is a connexion between the example above and some old results
of[KT] concerning M-algebras in various contexts, proved under mild conditions onV— namely, right closedness and the existence ofpullbacks. First, it was shown in [KT, Prop. 23.2] that, if M ∈ MonV is the free monoid on X ∈ V, then f M (or more properly (MonF)M) is the free monad on the endfunctor X⊗ −; secondly, it was shown in [KT, Prop. 28.2] that MonF preserves such colimits as exist. Ofcourse these are trivial consequences ofthe existence ofthe right adjoints in Corollary 3.2, when V satisfies the stronger conditions ofthis corollary. It is also observed in [KT, Section 23.2] that each of the results above fails when V is the non-right-closed monoidal category (Set, +,0).
Again, we may single out the simple case where V is the categorySet ofsmall sets:
3.4. Corollary. Let the locally-small categoryA admit small multiples and small
pow-ers and have a small generating set; then the functor f : Set −→[A,A] sending the set X to the endofunctor X · − (where X·A denotes the X–fold multiple of A) has a right adjoint g, which underlies a right adjoint to the functor Mon −→ MndA sending the monoidM to the monad M· − on A; here Mon = Mon (Set) is the category of monoids in the usual sense.
value ofa V-functor Vop⊗Aop⊗A−→V, so the representing object X∗A on the left side is the value ofa V-functor Ten : V⊗A−→A, whose underlying ordinary functor is ofcourse the ∗ : V × A−→ A corresponding to f : V−→[A,A]. In particular, each f X =X∗ − in [A,A] underlies a V-functor Ten(X,−) :A−→A, and each f x=x∗ −: (X∗ −)−→(X′∗ −) induced by x :X−→X′ in V underlies a V-natural transformation
Ten(x,−) : Ten(X,−)−→ Ten(X′,−). That is to say, f : V−→[A,A] factorizes as a
composite
V q //(A,A)
u //[A,A], (3.8)
where (A,A) is the (ordinary) category of V-functors A−→A and V-natural transfor-mations between these (as distinct from theV-category [A,A] of[KB, Section 2.2], which rarely exists whenA is large), andu is the functor sending theV-functorH :A−→A to its underlying ordinary functoruH =H0 :A−→ Aand sending theV-naturalα :H−→K to the underlying natural transformation α0 : H0−→K0; here, ofcourse, we have qX = Ten(X,−) and qx = Ten(x,−). Since α0 has the same components (α0)A = αA as α,
the functor u is faithful; recall from [KB, Section 1.3] thatu is fully faithful whenI is a generator of V. Indeed still more is true : (3.8) underlies a factorization
V
Q=(q,q,q ◦)//(A,A)U=(u,u,u ◦)//[A,A] (3.9)
ofthe strong monoidal functor F into strong monoidal functors Q and U : here qxy is
the isomorphism Ten(X,Ten(Y,−)) ∼= Ten(X ⊗Y,−) and q◦ is the isomorphism 1 ∼=
Ten(I,−), while uHK is the isomorphism K0H0 ∼= (KH)0 and u◦ is the isomorphism
1A = (1A)0.
3.5. Theorem. Let the symmetric monoidal closed V and its action F :V−→ [A, A] on A satisfy the hypotheses of Theorem 3.1, and let A be the corresponding V-category as in that theorem. Then q has a right adjoint r, so that Q has in MON CAT a right adjoint R, and MonQ: MonV −→ Mon(A,A) has a right adjoint MonR.
Proof. As in the proofofTheorem 3.1, we use the Special Adjoint Functor Theorem.
We still have on V the hypothesis (i)—(iii) above, and q like f preserves small colimits : for one easily sees that such colimits in (A,A), like those in [A,A], are formed pointwise from those in A. Moreover each (A,A)(qX, H) is a small set, being in effect a subset of [A,A](uqX, uH) = [A,A](f X, uH) ∼=V(X, guH).
3.6. Remark. In the situation ofTheorem 3.5, as a mate ofthe equality f = uq we
have the canonical comparison ζ : r−→gu, as well as such a comparison ξ : R−→GU and the resulting Monξ: MonR−→(MonG)(MonU). For aV-functor H:A−→A, the comparison morphism
ζH :rH−→guH (3.10)
transformations X ∗A−→H0A. Since V-naturality reduces to mere naturality when I is a generator of V, we have rH = guH in this case, for a V-functor H : A−→A. (Of course this does not reducegtor: f orgT is defined even for a mere functorT :A−→ A.) Note that we can make similar comments about ξH, when now H is a V-monad on A;
for then (ξH)A = (ζH)A for eachA∈A. In particular we have such results in the special
case where AisV with the canonical action, so thatA isV. As for the other special case given by V =Set, there is here no difference between A and A, or between r and g.
4. An explicit formula for the right adjoints
g
and
r
.
We continue to suppose thatF = (f,f , f ◦) :V−→[A,A], corresponding to∗:V× A−→ A
along with α and λ, is an action of V on A for which we have the adjunction (2.1). For X ∈ V and T : A−→ A, the set [A,A](f X, T) = [A,A](X ∗ −, T) is the set ofnatural transformations (βA :X∗A−→T A), which is isomorphic to the set ofnatural
transfor-mations (γA : X−→A(A, T A))—now in the generalized sense of[EC]; and to say that
this set admits a representation ofthe form V(X, Y) is, by the definition ofan end, to say that the end A∈AA(A, T A) exists. In other words:
4.1. Proposition. For an action F = (f,f , f ◦) of V on A, the functorf :V−→[A,A]
has a right adjoint g if and only if, for each endofunctorT of A, the end A∈AA(A, T A) exists; whereupon we have
gT =
A∈A
A(A, T A). (4.1)
4.2. Remark. This is a large end, in so far as the category A is seldom small in the
examples of interest : so that the existence of this end does not follow from completeness of V. The formula (4.1) does nothing to establish the existence of the right adjoint g, but merely gives an explicit formula for it when its existence is otherwise assured, as for instance by the hypotheses ofTheorem 3.1.
4.3. Remark. We observed in the Introduction that, whenf has a right adjointg, then
also F has a right adjoint G in MON CAT, so that MonF : MonV−→ MndA has a right adjoint MonG. This last, as we said, agrees with g on objects : one says loosely that a monoidal functor, such as G, “takes monoids to monoids”. It is easy to see how this goes in the present case, using the formula (4.1). Let T = (T, m, i) be a monad on
A, and write
τA:gT−→A(A, T A) (4.2)
for the counit of the end (4.1); that is to say, τ = (τA) is universal among natural
transformations into A(A, T A). Now the natural transformation given by the composite
I∗A λ
A
//A
iA
gives by the adjunction a natural transformation
jA:I−→A(A, T A), (4.4)
which factorizes asτAj for a unique morphism
j :I−→gT. (4.5)
Next, the composite
gT ⊗gT τ
T A⊗τA
//A(T A, T T A)⊗A(A, T A)
M //A(A, T T A)A(A,mA)
//A(A, T A) (4.6)
is natural in A by the Eilenberg-Kelly calculus of[EC], so that it factorizes as τAn for a
unique morphism
n :gT ⊗gT−→gT; (4.7)
and it is (gT, n, j) that is the monoid (MonG)T ofMonV.
There is a similar formula for the right adjoint r of q : V−→ (A,A), where now V
is symmetric monoidal closed; once again, it involves a large end, and thus is ofvalue only when r is independenly known to exist, as under the hypotheses ofTheorem 3.5. Here (A,A)(qX, H) = (A,A)(Ten(X,−), H) is the set of V-natural transformations (ρA: Ten(X, A)−→HA), which is isomorphic to the set of V-natural transformations
(σA:X−→A(A, HA)). Accordingly—see Section 2.2 of[KB]—we have:
4.4. Proposition. In the situation of Theorem 3.5, for a V-functor H : A−→A we
have
rH =
A∈A
A(A, HA) ; (4.8)
moreover, when H = (H, m, i)is a V-monad on A, we find(MonR)H as the object (4.8) of V with a monoid-structure formed as in Remark 4.3.
4.5. Remark. As in Section 2.2 of[KB], we may write (4.8) in the form
rH = [A,A](1, H), (4.9)
even though, for large A, the functor category [A,A] may not exist as a V-category; the point is that the particular hom-object (4.9) does exist in V under the hypotheses of Theorem 3.5. Note the special case where H is the identity functor 1 = 1A : A−→A;
here (4.9) becomes
r1A = [A,A](1,1) =
A∈A
which is the monoid inV usually called thecentre oftheV-categoryA. In the case where
A is V itselfwith the canonical action, so that A = V, the V-functor 1V : V−→V is
represented by the objectI ofV, so that the enriched Yoneda lemma allows us to rewrite (4.9) as
rH =HI for H :V−→V ; (4.11)
in particular we have
r1V = I . (4.12)
The canonical comparison ζ :r−→gu ofRemark 3.6 has forζH the monomorphism
[A,A](1, H) =
A∈A
A(A, HA)−→
A∈A0=A
A(A, HA) ; (4.13)
we observed that this is invertible whenI is a generator of V, so thatV-naturality reduces to mere naturality. To see that it is not invertible in general, even when A = V for a well-behaved V, take V to be the symmetric monoidal closed category ofgraded abelian groups, and take H to be 1V. Then the domain of(4.13) is I by (4.12), so that (4.13)
takes the form
ζ :I−→
A∈V
[A, A]. (4.14)
To see that (4.14) is not invertible in V, consider its image under the functor V(I,−) :
V−→ Set. Ofcourse we have I0 =Z and In = 0 for n = 0; so a morphism f : I−→I
in V has fn = 0 for n = 0 while f0 : Z−→Z is multiplication by n for some n ∈ Z; in
this way, we haveV(I, I)∼=Z. On the other hand, to give an element ofV(I,A∈V[A, A]) is to give a family (αA: I−→[A, A])A∈V which is natural in A; equivalently, it is to give
a family (βA : A ∼= I ⊗A−→A) that is natural in A, or an element β ofthe centre of
the ordinary category V. For any sequence (kn)n∈Z ofintegers, we obtain such a family
by setting (βA)n(a) = kna for a ∈ An; and consideration ofthose A having for some m
the components An = 0 for n = m and Am =Z shows that every natural family (βA) is
ofthis kind. Thus the monoid V(I,A∈V[A, A]) is the power ZZ of Z, and (4.14) is not invertible.
4.6. Remark. When V = Set, there is, as we said, no difference between A and A or
between r and g, and for any T :A−→ Awe have the formula
gT =
A∈A
A(A, T A) = [A,A](1, T) ; (4.15)
in particularg1A is thecentre [A,A](1,1) ofthe categoryA. In the still more special case
the formula (4.15) for the right adjointg : [Set, Set]−→ Setreduces, since 1 : Set −→
Set is represented by 1, to
gT = T1. (4.16)
Note what this becomes when we pass to the monoids and consider the functor MonF : Mon−→ Mnd(Set) sending the monoidM to the monadM×−onSet; the right adjoint MonG: Mnd(Set)−→Mon sends the monadT = (T, m, i) on Set, which we may think ofas a (perhaps infinitary) Lawvere theory, to the monoid T1 ofits unary operations.
4.7. Remark. Consider again, in the case V = Set, the functor MonF : Mon−→
MndA; we can compose it with the inclusion functorm: Grp−→ Mon, which considers each group as a monoid, to obtain a functor h = (MonF)m : Grp−→ MndA whose domain is the category Grp ofgroups. Since MonF has a right adjoint MonG when
A satisfies the conditions ofCorollary 3.4, and since m has a right adjoint n sending a monoid to the group ofits invertible elements, sohtoo has a right adjointk=n(MonG). By (4.15), its explicit value at a monad T on A is the group ofinvertible elements in the monoid [A,A](1, T); in the case A = Set, this is the group ofinvertible elements of T1. For a group X, observe that hX is the monad X · − on A. The case A = Grp is a familiar one : an (X · −)-algebra is a group A along with a group homomorphism X−→ AutA, or equally an actiona:X×A−→AofX on the groupA. As is well known, the category ofsuch algebras is isomorphic to that ofpairs (u:B−→X, v :X−→B) of group-homomorphisms with uv = 1; here B is the semi-direct product of X with (A, a), while conversely A is the kernel ofu with the appropriate action.
4.8. Remark. Consider again the case A=V, and suppose again for simplicity that V
is symmetric monoidal closed; so thatf :V−→[V,V] factorizes as
V q //(V,V)
u //[V,V], (4.17)
and MonF factorizes as
MonV
MonQ //(V-Mnd)V MonU //MndV, (4.18)
where (V-Mnd)V is the category of V-monads on V and V-natural monad maps. The functor q is fully faithful; for a V-natural family (βZ :X⊗Z−→Y ⊗Z) corresponds by
the adjunction (2.2) to a V-natural family (γZ :Z−→[X, Y ⊗Z]), which in turn (since
we have the representation Z ∼= [I, Z] of the V-functor 1V :V−→V) corresponds by the
enriched Yoneda lemma to a morphismb :X−→Y ofV, so thatβZ =b⊗Z for this unique
b. In fact MonQ too is fully faithful; for, whenX andY are monoids, b is a monoid map when the βZ =b⊗Z constitute a monad map, as we see on taking Z =I. We observed
4.9. Example. Even for a symmetric monoidal closed category V that is locally finitely
presentable (both as a category, and as a closed category in the sense of[KA]), neither f : V−→[V,V] nor MonF : MonV−→ MndV need be full. It suffices to prove the second ofthese : for MonF is full when f is so, since if X and Y are monoids in V and z :X−→Y inV is such that z⊗ −: (X⊗ −)−→ (Y ⊗ −) is a monad map, taking the I-componentz⊗I :X⊗I−→Y ⊗I showsz itselfto be a monoid map. For our counter-example we take V to be the symmetric monoidal closed category ofdifferential graded abelian groups—that is, ofchain complexes ofabelian groups—and chain maps. Here I is the chain complex given by I0 = Z, In = 0 for n = 0; and f I = I⊗ – is (to within
isomorphism) the identity endofunctor 1V of V. In f act I is trivially a monoid in V, and
(MonF)I ∼= 1V is the identity monad on V. Now write J for the chain complex having
J−1 = Z and Jn = 0 for n = −1, with ofcourse the zero boundary map; and observe
that DA =J ⊗A is the desuspension of A, given by (DA)n = An+1, the boundary map
(DA)n−→(DA)n−1 being the negative ofthe boundary map An+1−→An. An algebra
for the endofunctor D of V is a pair (A, a) where A∈ V and a:DA−→A; these form a categoryD-Alg with a forgetful functorUD :D-Alg−→ V. Write Φ :V−→D-Alg for the
functor sending A ∈ V to the D-algebra (A, a), where a :DA−→A has the components an : (DA)n−→An given by the boundary map An+1−→An of A; clearly a is indeed a
chain map, and Φ is a functor with UDΦ = 1 :V−→ V. Let z :J−→M exhibit M as the
free monoid in V on the object J of V; we don’t really need the explicit value ofM, but ofcourse M =n≥0J⊗n has M
n =Z for n ≤0 and Mn= 0 for n > 0, with zero as the
boundary map of M. BecauseV is right closed, it follows from Proposition 23.2 of [KT] that z induces an isomorphism, commuting with the forgetful functors,
(z⊗ −)∗ : (M ⊗ −)-Alg −→(J ⊗ −)-Alg =D-Alg, (4.19)
where (M⊗ −)-Alg is the category ofalgebras for the monad M⊗ − , while (J⊗ −)-Alg is the category of algebras for the mere endofunctor J ⊗ − = D. So there is a functor Ψ :V−→(M⊗ −)-Alg, commuting with the forgetful functors, for which (z⊗ −)∗Ψ = Φ.
Now V is the category 1V-Alg ofalgebras for the identity monad 1V ∼= I ⊗ − on V, so
that Ψ is a functor (I⊗ −)-Alg−→(M⊗ −)-Alg commuting with the forgetful functors; therefore, by a classical result (see for example Proposition 3.4 of [KS]), Ψ = θ∗ for a
unique monad map θ : (M ⊗ −)−→(I⊗ −). IfMonF were full, θ would be w ⊗ – f or some monad map w : M−→I; which, since M is the free monoid on J, corresponds to some mere morphism t=wz :J−→I. Now we have
Φ = (z⊗ −)∗Ψ = (z⊗ −)∗θ∗ = (z⊗ −)∗(w⊗ −)∗ = (t⊗1)∗. (4.20)
But this cannot be so : the only morphism t:J−→I in V is the zero morphism, so that (t⊗ −)∗Afor A∈ V is (A, 0 :DA−→A), which is not equal to ΦAin general. So MonF
5. Alternative expressions for
g
and Mon
G
as large limits.
It turns out that, for an action F : V−→[A,A] admitting the adjunction (2.1), one can easily describe gT for certain special endofunctors T of A; we now show that each endofunctor T is canonically a limit ofthese special ones, so thatgT, when it exists, may be expressed as a limit ofthe gP for the special P. Once again, since the limit involved is a large one, this expression forgT does nothing to establish its existence, for which we must refer to such results as Theorem 3.1. Note that, although (MonG)T for a monad T is just gT with the appropriate monoid-structure, we elect also to give below a second limit-formula for (MonG)T, which makes sense only whenT is a monad, but then gives a tidier result, expressing (MonG)T as a “smaller” large limit. So each ofour considerations in this section will have two forms : one for endofunctors and one for monads.
First, recall again the classical result used in the last section, asserting that, for monads T and S on a category A, every functor T-Alg −→S-Alg commuting with the forgetful functors to A is ofthe form θ∗ for a unique monad map θ : S−→T. We now give an
analogue ofthis in which endofunctors ofA replace monads.
If T is a mere endofunctor of A it is usual to employ the name T-algebra, as we did in Example 4.9 above, for a pair (A, a) where A ∈ A and a : T A−→A is any morphism in A. Such T-algebras do not serve our purpose here : rather we define a T -quasi-algebra to be a triple (A, a, C) where A, C ∈ A and a : T A−→C is a morphism in A
(called the quasi-action). Then a map (A, a, C)−→(B, b, D) of T-quasi-algebras is a pair (u:A−→B, v:C−→D) making commutative the diagram
T A a //
T u
C
v
T B b //D ;
so thatT-quasi-algebras and their maps form a categoryT-QAlg, with a forgetful functor to A2 = A × A. Ofcourse, T-QAlg is just the comma category T /1; but our present notation is chosen to emphasize the analogy between it and T-Alg. For any morphism θ : S−→T of endofunctors, we obtain a functor θ† : T-QAlg −→ S-QAlg, commuting
with the forgetful functors to A2 : it sends the T-quasi-algebra (A, a : T A−→C, C) to the S-quasi-algebra (A, a.θA : SA−→C, C) whose quasi-action is the composite of
θA:SA−→T A and a:T A−→C.
5.1. Lemma. Every functor Φ: T-QAlg−→ S-QAlg commuting with the forgetful
func-tors is of the form θ† for a unique morphism θ:S−→T in [A,A].
Proof. Given Φ, we define θA by setting
(A, θA:SA−→T A, T A) = Φ(A,1T A :T A−→T A, T A);
then θ is a natural transformation because Φ sends the morphism
giving T u.θA = θB.Su. If now (A, a :T A−→C, C) is a general T-quasi-algebra, sent by
Φ to (A, a:SA−→C, C), then Φ sends the morphism
(1A, a) : (A,1, T A)−→(A, a, C) to (1A, a) : (A, θA, T A)−→(A, a, C),
giving a=a.θA, as desired.
To introduce the “special” endofunctors and “special” monads we have referred to, we begin by recalling some classical actions. First, for each category A, we have the action ofthe monoidal [A,A] onA corresponding to the identity functor [A,A]−→[A,A]; in its “functor of two variables” form it is the evaluation functor E : [A,A]× A−→ A sending (T, A) to T A; the partial functor E(−, A) is what we earlier called e(A) : [A,A]−→ A. Provided that, as we henceforth suppose, A is complete and locally small, this action admits an adjunction ofthe form (2.1), namely
A(T A, B) ∼= [A,A](T,A, B), (5.1) and ifthe morphism a :T A−→A corresponds under the adjunction (5.1) to the natural transformation α : T−→ A, A, then one easily verifies that a is an action of T on A precisely when α is a map of monads.
for to give a morphism γ : T−→ {u, v} is to give morphisms α : T−→ A, C and β :
Moreover, just as each A(A, A) in the context ofthe adjunction (2.1) is a monoid in
V, so here each {u, u} is a monad on A, and for a general monad T on A, a morphism
is an action ofthe monad T onu—which is equally to say that aand b are actions ofthe monad T on A and on B for which the square above commutes (that is, for which u is a map of T-algebras). To give such a monad map γ : T−→ {u, u} is, ofcourse, equally to give monad maps α : T−→ A, A and β : T−→ B, B for which A, uα = u, Bβ; whereupon α = σu,uγ and β = τu,uγ. Ifhere we take in particular T to be {u, u} and γ
the identity, we see that σu,u and τu,u are monad maps in the special case of(5.3) given
by
We shall now show that an arbitrary endofunctorT ofAis canonically a limit in [A,A] ofendofunctors ofthe special forms A, C and {u, v}; and that an arbitrary monad T on A is canonically a limit in MndA ofmonads ofthe special forms A, A and {u, u}. (The referee has kindly pointed out that, since {1A,1C} =A, C, every endofunctor (or
ofthese limit-diagrams, we need to recall the notion ofaderived category (in the sense of Kan).
To each category K we associate a derived category K′; namely the free category on
the following graph (which we may also, loosely, callK′). For each objectK of K, there is
an object K′ of K′, and for each morphism u:K−→L of K there is an object (K, u, L)′
of K′, which might be written just as u′ ifit is agreed that a morphism determines its
domain and codomain; there are no other objects of K′. For each morphism u:K−→L
of K, there are edges su : (K, u, L)′−→K′ and tu : (K, u, L)′−→L′ ofthe graph K′; and
there are no other edges ofthis graph. The free category K′ generated by this graph
has ofcourse as morphisms, besides these edges, in addition the identity morphisms 1K′
and 1u′—but that is all. (Note that an identity morphism 1K : K−→K in K just gives
rise, like any other morphism, to an object (K,1K, K)′ of K′ and the associated edges
s1K, t1K : (K,1K, K)
′−→K′ of K′.) Ofcourse K′ is large whenever K is so.
For each endofunctor T of A (still supposed to be complete and locally small) we define as follows a functor
T : (T-QAlg )′−→[A,A]. (5.5)
For aT-quasi-algebra (A, a, C) we takeT(A, a, C)′ to be the endofunctorA, Cof A; f or
a map (u, v) : (A, a, C)−→(B, b, D) of T-quasi-algebras we take
T((A, a, C),(u, v),(B, b, D))′
to be the endofunctor {u, v}of A, and we askT to send
s(u,v) : ((A, a, C),(u, v),(B, b, D))′−→(A, a, C)′
and
t(u,v) : ((A, a, C),(u, v),(B, b, D))′−→(B, b, D)′
to the natural transformationsσu,v andτu,v of(5.3). Next, we describe a coneφT in [A,A],
with vertex T, over the functor T of(5.5). For its component φT
(A,a,C)′ : T−→ A, C we
take the natural transformation α corresponding to the quasi-action a : T A−→C; and for its component φT
((A,a,C),(u,v),(B,b,D))′ : T−→ {u, v} we take the natural transformation
γ corresponding as above to the morphism (u, v) : (A, a, C)−→(B, b, D) of T -quasi-algebras. Clearly φT is a cone, since σ
u,vφT(u,v)′ =φ
T
(A,c,C)′ and τu,vφ
T
(u,v)′ =φ
T
(B,b,D)′.
Alongside the functor T of(5.5) and the cone φT ofvertex T over T, we have the
following analogue adapted to monads : for each monad T on A we define as f ollows a functor
T : (T-Alg)′ −→ MndA. (5.6)
For a T-algebra (A, a) = (A, a : T A−→A) we take T(A, a)′ to be the monad A, A
the monad {u, u} on A; and we ask T to send su : ((A, a), u,(B, b))′−→(A, a)′ and
tu : ((A, a), u,(B, b))′−→(B, b)′ to the monad maps σu,u and τu,u of(5.4). Then we
describe a coneψT in MndA, with vertex T, over the functor T of(5.6) : its component
ψT
(A,a)′ : T−→ A, A is the monad map α corresponding to the action a :T A−→A, and
its component ψ((A,a),u,(B,b))′ : T−→ {u, u} is the monad map γ corresponding as above
to the map u : (A, a)−→(B, b) of T-algebras; once again, ψT is clearly a cone, having
σu,uψuT′ =ψT(A,a)′, and τu,uψ
T
u′ =ψ(TB,b)′.
The following observations seem to be new:
5.2. Proposition. For a complete and locally-small category A, the cone φT is a limit
cone in[A, A]for each endofunctor T of A, and the cone ψT is a limit cone inMndA for
each monad T on A. Thus each endofunctor of A is canonically a limit of those of the special forms A, C and {u, v}, while each monad on A is canonically a limit of those of the special forms A, A and {u, u}.
Proof. We prove only the monad version : the proof for the endofunctor version follows
exactly the same pattern, using Lemma 5.1 in place ofthe classical result about monad maps. So let T be a monad on A, and consider a cone ρ in MndA over the functor
T : (T-Alg)′ −→ MndA, the vertex ofthis cone being the monadS. For each T-algebra
(A, a), the component ρ(A,a)′ :S−→ A, Aof ρ corresponds to an action a:SA−→A of
the monad S on A, and hence to an S-algebra (A, a). For each map u : (A, a)−→(B, b) of T-algebras, the component ρu′ :S−→ {u, u} of ρ corresponds to an action
SA a //
Su
A
u
SB
b
//B
ofthe monad S on u. Since, however, the components of ρ satisfy σu,uρu′ = ρ(A,a)′ and
τu,uρu′ = ρ(B,b)′, the a and the b above are in fact the a and the b ofthe S-algebras
(A, a) and (B, b). So (A, a) −→(A, a) gives a functor T-Alg−→S-Alg commuting with the forgetful functors to A, which therefore has the form θ∗ for a unique monad map
θ :S−→T. Equivalently, the cone ρ has the form ψTθ for a unique θ :S−→T, showing
ψT to be a limit cone.
Still with A complete and locally small, let us once again consider an action F : V −→[A,A] ofa monoidalV onA, which we further suppose to admit an adjunction (2.1). Composing F with the action [A,A]−→[A2
,A2
] above gives an action V−→[A2
,A2
], which in its form V × A2
−→ A2
sends (X, u : A−→B) to (X ∗u : X ∗A−→X ∗B). Provided thatV has pullbacks, this too admits an adjunction ofthe form (2.1), namely
A2
where A(u, v) is given by the pullback
in V; note that (5.2) and (5.3) are just the special cases of(5.7) and (5.8) obtained by taking V to be [A,A] and F :V−→[A,A] to be the identity.
A right adjointg tof :V−→[A,A] certainly exists locally at the endofunctor A, C; for (5.1) and (2.1) give us, naturally inX, the composite isomorphism
[A,A](f X,A, C)∼=A((f X)A, C) =A(X∗A, C)∼=V(X,A(A, C)), (5.9)
showing that we can take
gA, C=A(A, C) (5.10)
as the local value ofthe right adjoint; the counitf gA, C−→ A, Cofthis representation (5.7) is ofcourse the morphismfA(A, C)−→ A, Ccorresponding to ǫ: (fA(A, C))A= A(A, C)∗A−→C.
The right adjoint g also exists locally at the endomorphism{u, v}; for (5.2) and (5.7) give us, naturally in X, the composite isomorphism
[A,A](f X,{u, v})∼=A2
((f X)u, v)) =A2
(X∗u, v)∼=V(X,A(u, v)), (5.11)
showing that we can take
g{u, v}=A(u, v) (5.12)
as the local value ofthe right adjoint, with the evident value ofthe counit f g{u, v} −→ {u, v}which we leave the reader to make explicit. On those endofunctorsT for which gT exists, g is of course functorial—it is a “partial functor”. In particular, one easily sees that the partial functor g carries the diagram (5.3) into the diagram (5.8).
5.3. Theorem. Let the action F :V−→ [A,A] of the monoidal V, which we suppose to
admit pullbacks, on the complete and locally small A admit an adjunction (2.1); then a right adjoint g of f is given locally at an endofunctor T of A by
gT = lim(T∗ : (T-QAlg)′ −→ V),
either side existing if the other does.
AlthoughgT has ofcourse a monoid structure whenT is a monad, there is a more direct alternative formula in the monad case, the domain of the limit now being the “simpler” (although still large) category (T-Alg)′. For when C =A in (5.9) and X is a monoid, an
elementαof[A,A](f X,A, A) is a monad map precisely when the corresponding element a ofA((f X)A, A) = A(X∗A, A) is an action, and thus precisely when the corresponding element κ(a) of V(X,A(A, A)) is a monoid map; so that (5.10) becomes an equality
(MonG)A, A = A(A, A)
ofmonoids, and similarly (5.12) becomes for v =u an equality ofmonoids
(MonG){u, u}=A(u, u).
In fact the left half of (5.8) becomes a diagram
A(u, u)
σu,u
??
τu,u
? ? ? ? ? ? ? ?
A(A, A)
A(B, B)
(5.13)
ofmonoids, describing a functor
T∗ : (T-Alg)′−→ MonV .
Now the corresponding argument gives :
5.4. Theorem. With the same hypothesis as in Theorem 5.3, a right adjoint MonG to
MonF is given locally at a monad T on A by
(MonG)T = lim(T∗ : (T-Alg)′−→ MonV),
5.5. Remark. We saw in Example 4.9 that, even for a locally-finitely-presentable
sym-metric monoidal closedV, when we consider the canonical actionF :V−→[V,V] of V on itself, neither f nor MonF need be fully faithful. Equivalently, the unit X−→g(X⊗ −) ofthe adjunction f ⊣ g is not in general invertible, even when X is a monoid in V. It follows that there is no analogue of Proposition 5.2 in which [A,A] is replaced by a general monoidal category V, even a well-behaved one. That is to say, a monoid X in V is not the (X-Alg)′–indexed limit ofthe special monoids ofthe forms V(A, A)(= [A, A]) and
V(u, u); for, by Theorem 5.4, this limit is precisely g(X⊗ −).
5.6. Remark. We have not found limit-formulae for rH and (MonR)H analogous to
Theorem 5.3 and 5.4. Certainly the action u : (A,A)−→[A,A], which as a functor (A,A)× A−→ A sends (H, A) to HA, admits an adjunction ofthe form (2.1), namely
A(HA, B)∼= (A,A)(H,A, B), (5.14)
where the V-functor A, B : A−→A is the right Kan extension ofthe V-functor B :I−→A along the V-functor A:I−→A, withIbeing the unit V-category having one object ∗ and having I(∗,∗) = I; this right Kan extension can be described in terms of the cotensor product by A, BC =|A(C, A), B|, and there is a canonical comparison
A, B0−→ A, B. Moreover, for the composite action
(A,A)−→[A,A]−→[A2
,A2
]
we again have an adjunction ofthe form (2.1), namely
A2
(Hu, v)∼= (A,A)(H,{{u, v}}), (5.15)
where {{u, v}} is defined by a pullback like (5.3) but with A, C replacing A, C and so on; ofcourse there is induced a canonical comparison {{u, v}}0−→ {u, v}. Putting H =qX in (5.14) we see, using the analogue of(5.9), that
rA, C=A(A, C) ; (5.16)
and similarly (5.15) gives
r{{u, v}}=A(u, v). (5.17)
There is an evident functor (to take the monad-adapted case)
H : (H-Alg0)′−→(A,A),
sending (A, a)′ to A, A and ((A, a), u,(B, b))′ to {{u, u}}, and a cone over this in
Mon(A,A) = (V-Mnd)A with vertex H. However the argument ofProposition 5.2 no longer applies to prove this a limit-cone : for we only get mere naturality, where
6. Appendix on tensored
V
-categories
We justify here the claim made in Section 2 that, for a right-closed monoidalV, to give a categoryAand an action ofV onAadmitting the adjunction (2.1) is essentially the same thing as to give a tensored V-categoryA. In f act we showed in Section 2 exactly how to produce, from the action of V on A and the adjunction (2.1), the V-category A and the
V-natural isomorphism k : A(X∗A,−) −→[X,A(A,−)] of(2.5) which constitutes the tensoring ofA; as our next step, we establish some further properties of the isomorphism k = kXAB : A(X ∗A, B)−→[X,A(A, B)] so produced, which we saw to be natural in
each variable, as well as being V-natural inB. First consider the following two diagrams, which have the same top edge and the same bottom edge, and thus can be thought ofas being pasted together along these to form a circular cylinder : we omit, to save space, the names ofthe interior objects, replacing each by the symbol @—the reader will have no trouble reconstructing them, guided by the names ofthe morphisms and our remarks below. (The arguments which follow were first used in [KC].) The diagrams are:
A(Z ∗(Y ∗(X∗A)), B) κ //
and α becomes a. However the left edges of (6.1) and (6.2) coincide by (1.1), so that the right edges coincide too; whence, by the Yoneda lemma, we have commutativity in the diagram
Now consider similarly the following two commutative diagrams:
A(Y ∗A, B) κ //
where again the two quadrangles commute by naturality, and the pentagon is an instance of(2.6); it is classical that the triangle commutes,ihere being an instance ofthe canonical isomorphism i:X−→[I, X]. Now the commutativity of(1.3) gives that ofthe diagram
Finally, consider the two diagrams of θ got by taking A to be V, and where the pentagonal region in (6.7) is an instance of(2.6) while the rectangle in (6.8) commutes by naturality. Here the commutativity of (1.2) gives
V(I, k)θ =φκ; (6.9)
in other words, ifwe useθandφto identifyAwithA0andVwithV0, thenκ:A(X∗A, B)
−→ V(X,A(A, B)) is the bijection V(I, k) underlying the isomorphism k : A(X∗A, B)
−→[X,A(A, B)] in V—which we may also express by saying that k is a lifting of the bijection κto an isomorphism inV. In the light ofthis, we may describe (6.3) as a lifting fromSET toV of(2.6), and similarly describe (6.6) as a lifting of(2.4). Note in particular the special case of(6.9) obtained when A=V, namely
V(I, p)φ=φκ . (6.10)
Now we may describe the association, for a right-closed monoidalV, between actions of
V admitting an adjunction (2.1) and tensoredV-categories. For the first ofthese, the data consist ofthe categoryA, the functor∗, the natural isomorphismsαandλsatisfying (1.1) and (1.3), the functor A:Aop× A−→ V (which we may write asA(−,−), to distinguish it from the V-category A), and the natural isomorphism κ of(2.1). The data for the second consist ofa V-category A and, for each pair X, A, an object X∗A of A together with a V-natural isomorphism
as in (2.8). The reader will recall from Section 2 the process—let us call itξ—leading from the first to the second : we construct the V-category A with obA = obA by giving it the hom-objects A(A, B) from the adjunction κ of(2.1), with the composition operation M given as the mate under κ ofthe composite ǫBC(1∗ǫAB)α of(2.3), and with the
unit operation j : I−→A(A, A) which is the image under κ of λ; then we use (2.6) to define kXAB, whose V-naturality in B follows from Lemma 2.1, after our checking that
the functor A:Aop× A−→ V does coincide, to within the isomorphisms (2.4), with the functor homA.
We now describe a process—let us call it η—going in the other direction. We take for the category Athe underlying categoryA0 ofthe givenV-categoryA, so that A(A, B) =
V(I,A(A, B)), and we use A : Aop × A−→ V as another name for the functor hom
A :
Aop0 ×A0−→ V arising from theV-categoryA. We take f or κ the composite bijection
A(X∗A, B) =V(I,A(X∗A, B))
V(I,k)//V(I,[X,A(A, B)]) φ−1 //V(X,A(A, B)), (6.11)
whereφ (as in (6.8)) denotes the isomorphism πV(ℓ,1), which is the caseA=V of(2.4); thus, modulo the isomorphism φ, the bijection κ is that underlying the isomorphism k of V. Since k is V-natural in the variable B, it is certainly natural in B, so that the same is true of κ; accordinglyX∗A, so far defined only on objects, extends to a functor
∗ : V × A−→ A in a unique way that makes κ = φ−1V(I, k) natural in X and A as well as in B. For α we take the unique morphism—clearly invertible—that makes (2.6) commute for each B; and for λ we take the unique morphism—again clearly invertible— that makes (2.4) the identity, so thatκA(λ,1) = 1. It remains to see thatαand λsatisfy the coherence conditions (1.1) and (1.3). Consider first the diagram (6.3) : because each arrow therein is V-natural in B, each leg has the form of a V-natural transformation ρ : A(K,−)−→T where K = Y ∗(X ∗A) and T = [Y,[X,A(A,−)]]. By the enriched Yoneda lemma, two such V-natural transformation ρ and ρ′ coincide ifand only if
I j
K
//A(K, K)
ρK
//T K = I
jK
//A(K, K)
ρ′
K
//T K . (6.12)
On the other hand, we have the ordinary natural transformations
V(I, ρ),V(I, ρ′) :A(K,−) =V(I,A(K,−))−→ V(I, T K);
and by the ordinary Yoneda Lemma, these coincide ifand only ifV(I, ρK)1K =V(I, ρ′K)1K;
which is the same criterion as (6.12). That is to say,ρ=ρ′ifand only ifV(I, ρ) =V(I, ρ′).
We conclude that (6.3) commutes ifits image under V(I,−) does so; but by (6.10) and (6.11), this image is essentially the commutative diagram (2.6). So (6.3) does com-mute; and now we can read (6.1) and (6.2) in the reverse direction, to conclude that (1.1) commutes. Similarly, each morphism in (6.6) is V-natural in B, so that (6.6) commutes ifits image under V(I,−) does so; but since A(A, B) = V(I,A(A, B)), the commutativity ofthis image reduces, using (6.11), to the trivially-verified fact that
the reverse order the diagrams (6.7) and (6.8), in which θ is here an identity, gives the desired commutativity of(1.3).
Suppose now that, starting with the data (A,∗, α, λ,A(−,−), κ), we first apply the process ξ to arrive at the data (A, k), and then the process η to arrive at the data (A′,∗′, α′, λ′,A′(−,−), κ′). We observed already in Section 2 that we have an isomorphism
θ = κA(λ,1) : A−→A0 = A′ given by (2.4), and that the functor A′(−,−) = homA
agrees with A(−,−) modulo θ. Since κ′ :A′(X∗A, B)∼=V(X,A(A, B)) isφ−1V(I, k) by (6.11), it follows from (6.9) that κ = κ′θ. Accordingly ∗′ agrees with ∗ modulo θ. Since
θ =κA(λ,1) whileλ′ is defined by κ′A′(λ′,1) = 1, we see thatλ′ agrees withλ moduloθ.
Finally, since k is defined in terms of α by (2.6), while α′ is defined in terms of k by the
primed version of(2.6), the observation above that κ = κ′θ implies that α′ agrees with
α modulo θ. Thus θ constitutes, in an evident sense, an isomorphism between the data (A,∗, α, λ,A(−,−), κ) and the image ofthis under the process ηξ.
Suppose on the other hand that, starting with the data (A, k), we first apply the process η to arrive at the data (A,∗, α, λ,A(−,−), κ), and then the process ξ to arrive at the data (A′′, k′′). Here obA′′ = obA = obA, and A′′(A, B) = A(−,−)(A, B) =
homA(A, B) = A(A, B). With κ defined as φ−1V(I, k) as in (6.11), recall that α is
defined in terms of k by (2.6), while k′′ is then defined in terms of α by (2.6); it follows
thatk′′ =k. Since λis so defined that theκA(λ,1) of(2.4) is the identity, the image ofλ
under κ:A(I∗A, A)−→ V(I,A(A, A)) is j :I−→A(A, A); but j′′ is this image, so that j′′ =j. We shall therefore have (A′′, k′′) = (A, k) ifwe can show that M′′ =M. Given
the definition of M′′ using the process ξ, we are to show, for any tensoredV-categoryA,
the commutativity ofthe diagram
(A(B, C)⊗A(A, B))∗A M∗1 //
α
A(A, C)∗A
ǫAC
A(B, C)∗(A(A, B))∗A)
1∗ǫAB
A(B, C)∗B ǫ
BC
//C ,
(6.13)
wherein theǫAB and so on are the counits ofthe adjunctionk, or equally ofthe adjunction
κ of(6.11). A simple way ofestablishing this commutativity is the following. When we see X ∗A as the value ofthe V-functor Ten : V ⊗A−→A, the isomorphism kXAB :
A(X∗A, B)−→[X,A(A, B)] is V-natural in each ofthe variables X, A, B : see Sections 1.10 and 1.11 of[KB]. By these same sections, the counit ǫAB : A(A, B)∗A−→B of
so too is α. Finally, M is V-natural in each variable by Section 1.8(g) of[KB]. All that we in fact use ofthe two legs of(6.13) is their V-naturality in the variable C. Using the V-adjunctions k and p, we can equally see these legs as two families of morphisms A(B, C)−→[A(A, B),A(A, C)], eachV-natural inC. By the enriched Yoneda lemma, the two legs coincide ifthey do so when we put C =B and compose with j :I−→A(B, B); but ifwe do this in (6.13) and use (1.2), each leg reduces toǫAB :A(A, B)∗A−→B. We
conclude that the process ξη is the identity : and this ends our appendix by providing the precise sense in which to give, for a right-closed monoidal V, a category A and an action of V on A admitting an adjunction (2.1) is essentially the same thing as to give a tensored V-categoryA.
References
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[EK] S. Eilenberg and G.M. Kelly, Closed categories, in Proc. Conf. on Categorical Alge-bra (La Jolla, 1965), Springer-Verlag, Berlin-Heidelberg-New York, 1966; 421-562.
[GP] R. Gordon and A.J. Power, Enrichment through variation, J. Pure Appl. Algebra 120 (1997), 167-185.
[KM] G.M. Kelly, On Mac Lane’s conditions for coherence of natural associativities, com-mutativities, etc., J. Algebra 1 (1964), 397-402.
[KC] G.M. Kelly, Tensor products in categories, J. Algebra 2 (1965), 15-37.
[KD] G.M. Kelly, Doctrinal Adjunction, inCategory Seminar, Sydney 1972-73, = Lecture Notes in Mathematics 420, Springer-Verlag, Berlin-Heidelberg-New York (1974); 257-280.
[KT] G.M. Kelly, A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on,Bull. Austral. Math. Soc.22 (1980), 1-83.
[KA] G.M. Kelly, Structures defined by finite limits in the enriched context I,Cahiers de Topologie et G´eom. Diff´erentielle 23 (1982), 3-42.
[KB] G.M. Kelly, Basic Concepts of Enriched Category Theory, London Math. Soc. Lec-ture Notes Series 64, Cambridge Univ. Press, 1982.
[ML] S. Mac Lane, Categories for the Working Mathematician, Springer-Verlag, New York-Heidelberg-Berlin, 1971.
G. Janelidze
Mathematical Institute of the Academy of Science M. Alexidze str. 1, 380093 Tbilisi, Georgia (Currently at Universidade de Aveiro, Portugal) G.M. Kelly
School of Mathematics and Statistics, F07 University of Sydney, N.S.W. 2006, Australia
Email: janelidze@mat.ua.pt maxk@maths.usyd.edu.au
